It has been theorized that a single simple rule generates all known complexity demonstrated by matter. Nevertheless, matter is currently described and depicted using many complex formulas. While drafting and geometrical computer programs have become commonplace, they rely on numerous complex formulas and require advanced computer processors and computer memory, and they cannot be used to model, truncate and create all “General Polyhedra”.
A simple computational method for modeling, truncating and creating “General Polyhedra” that allows one to model and study any form in n dimensions has been needed. N-dimensional forms are created from lines, polygons, and polyhedra. For example, a 2-dimensional polygon has 1-dimensional sides, a 3-dimensional polyhedron has sides or faces which are 2-dimensional polygons, a 4-dimensional polyhedron has sides that are 3-dimensional polyhedra, etc. A 4-dimensional polyhedron that has four sides, for example, could be constructed by building a 3-dimensional “core” polyhedron, and building four “side” 3-dimensional polyhedra that represent the sides of the 4-dimensional polyhedron. Each “side” 3-dimensional polyhedron has a face with dimensions that are mirror image with one face of the “core” 3-dimensional polyhedron, etc.
Old methods have named polyhedra and grouped them into special categories depending on their characteristics. A “General Polyhedron” may be referred to in this new method simply as “Polyhedron” with “Polyhedra” for plural, has n dimensions where n faces can have a common edge, can be concave or convex and can have holes or no holes, and can have symmetry or no symmetry. That represents all matter.
Old Art to model, truncate, and manipulate polyhedra has had limitations that do not generalize to all general polyhedra. First, the definition of polyhedra has been unsatisfactory. Polyhedra have been named and grouped into categories based on some common characteristics that they share. A defining characteristic of almost all kinds of polyhedra has been that just two faces join along any common edge. A polyhedron has been a 3-dimensional example of the more general polytope in any number of dimensions. Then, truncation of the polyhedron as defined in Math World has been the removal of portions of solids falling outside a set of symmetrically placed planes. Truncation displaces points along the edges of a polyhedron by a ratio r less or equal to ½, where r is the fraction of the edge length at which to truncate, and then constructing the new polygons. And rectification is the process of truncating a polytope by marking the midpoints of all its edges, and cutting off its vertices at those points.
Another prior art method, described in U.S. Pat. No. 5,428,717 to Glassner, can truncate only a simply connected concave object with no holes, by transforming it first to its convex hull by popping and sliding the concave edges, identifying midpoints of edges to be used for computations and then truncating the convex hull using computations of intersecting moving planes and half spaces inspired by the classical construction of dual polyhedra. The dual of a Platonic solid or Archamedean solid, for example, can be computed by connecting the midpoints of the sides surrounding each polyhedron vertex, and constructing the corresponding tangential polygon. Then the truncated convex hull will have to be converted again to the concave goal shape. The Glassner method requires extensive computation and has its obvious limitations.
Another prior art method, described in U.S. Pat. No. 6,396,494 to Pittet, frames the object to be studied with known geometric shapes of their choice like the tetrahedra that form a cube, then slices said geometric shapes using computations of intersecting planes. To speed up such very complex computations that require special quick hardware such as processors and computer systems, labor intensive information for a limited number of slices, 16 predetermined slices in this instance only, are computed by the operator first, then pre-entered into the system to be used, which leads to limited possibilities and missed information.
The invention disclosed herein uses simple mathematical computations of angles and side lengths of triangles to model and truncate any n-dimensional General Polyhedron, not limiting truncation up to the midpoint of edges of the polyhedron, and not using complex calculations of intersecting planes or half planes, or intersecting moving planes and half spaces. Using this new method, and changing the value of a few variables, one can quickly truncate any polyhedron along any edge by any proportion possible of that edge, and at any angle possible ad infinitum, without disregarding any information.
More particularly, using this novel simple computational method, one can model a Polyhedron. Then one can truncate or slice or clip any vertex of said polyhedron at any angle or in parallel, and along any proportion of each and every edge of said polyhedron that meet at said vertex from 0% and up to the maximum amount of truncation possible. Truncating said polyhedron would create new faces of the polyhedron that can be measured and built thus creating an infinite amount of new polyhedra. Then one can model another one or more new polyhedron with one face that has dimensions that are mirror image with either an old face of the polyhedron, or with a new face created by the truncation. Having the old and new polyhedron share a face will create one more new polyhedron. And so on and so forth, one can imagine the infinite complexity that can be generated and studied. For example, one can tessellate n-dimensional spaces by filling them with n-dimensional polyhedra.
One can also model a new polyhedron with all its faces having mirror image dimensions of the faces of the original polyhedron, thus being able to depict a R handed polyhedron and a L handed polyhedron for example, which will facilitate study.
Using this novel method, the polyhedron built will help us be able to model, understand and manipulate structures at the nano scale for the study of molecular biology for example, or bioinformatics, organic chemistry, crystallography etc; and structures at the macro scale to build objects that can be perceived by our senses; and structures at such large scale that our senses cannot perceive. And since physics describes matter as the geometric distribution of forces in space, then the study and use of this method that uses only angles and side lengths of triangles may help us understand these forces. The geometric distribution of forces also helps us understand the other structures in life, like the dynamics in a family or corporation. In a family, for example, the special bond between the father and daddy's little girl, the special bond between the mother and the son, and the special bond between the father and the mother, define a triangle of their own. Thus the triangle is the trinity that is the one that is the everything.
The present invention will become obvious after the following description.